We can bound the expectation term using Dudley entropy bound (Lemma A.5 of [1]}),i.e.
\(\nonumber &\mathrm {E}_{\sigma }\left[\max \limits _{\theta \in \Omega (\widehat{r},s)} \left|\frac{1}{m GF}\sum _{i=1}^m \sigma _i (f(x_i^a;\theta ) - y(x_i^a))\nabla f(x_i^a;\theta ) \right|\right] \\\nonumber \le & \mathrm {E}_{\sigma }\left[\max \limits _{\theta \in \Omega (\widehat{r},s), h \in S_d} \left|\frac{1}{m GF}\sum _{i=1}^m \sigma _i (f(x_i^a;\theta ) - y(x_i^a))\langle \nabla f(x_i^a;\theta ), h\rangle \right|\right] \\\le & \inf \limits _{\alpha } \left(\frac{4\alpha }{\sqrt{m}} + \frac{12}{m}\int _\alpha ^{\sqrt{m}}\sqrt{\log \mathcal {N}(\mathcal {G}(\widehat{r},s);\epsilon )} d\epsilon \right)\) 
